Embedded newform 2200.1.d.b.901.1

• Label: 2200.1.d.b.901.1
• Level: $2200$
• Weight: $1$
• Character: 2200.901
• Self dual: yes
• Analytic conductor: $1.098$
• Analytic rank: $0$
• Dimension: $1$
• Projective image: $D_{3}$
• CM discriminant: -88
• Inner twists: $2$

Newspace parameters

Level:	\( N \)	\(=\)	\( 2200 = 2^{3} \cdot 5^{2} \cdot 11 \)
Weight:	\( k \)	\(=\)	\( 1 \)
Character orbit:	\([\chi]\)	\(=\)	2200.d (of order \(2\), degree \(1\), minimal)

Newform invariants

Self dual:	yes
Analytic conductor:	\(1.09794302779\)
Analytic rank:	\(0\)
Dimension:	\(1\)
Coefficient field:	\(\mathbb{Q}\)
Coefficient ring:	\(\mathbb{Z}\)
Coefficient ring index:	\( 1 \)
Twist minimal:	yes
Projective image:	\(D_{3}\)
Projective field:	Galois closure of 3.1.2200.1
Artin image:	$D_6$
Artin field:	Galois closure of 6.2.24200000.1
Stark unit:	Root of $x^{6} - 486x^{5} + 15355x^{4} - 206460x^{3} + 15355x^{2} - 486x + 1$

Embedding invariants

Embedding label			901.1
Character	\(\chi\)	\(=\)	2200.901

$q$-expansion

\(f(q)\)

\(=\)

\(q-1.00000 q^{2} +1.00000 q^{4} -1.00000 q^{8} +1.00000 q^{9} +1.00000 q^{11} +1.00000 q^{13} +1.00000 q^{16} -1.00000 q^{18} -1.00000 q^{19} -1.00000 q^{22} +1.00000 q^{23} -1.00000 q^{26} -1.00000 q^{29} -1.00000 q^{31} -1.00000 q^{32} +1.00000 q^{36} +1.00000 q^{38} +1.00000 q^{43} +1.00000 q^{44} -1.00000 q^{46} -2.00000 q^{47} +1.00000 q^{49} +1.00000 q^{52} +1.00000 q^{58} +2.00000 q^{61} +1.00000 q^{62} +1.00000 q^{64} -1.00000 q^{71} -1.00000 q^{72} -1.00000 q^{76} +1.00000 q^{81} +1.00000 q^{83} -1.00000 q^{86} -1.00000 q^{88} -1.00000 q^{89} +1.00000 q^{92} +2.00000 q^{94} +1.00000 q^{97} -1.00000 q^{98} +1.00000 q^{99} +O(q^{100})\)

Character values

We give the values of \(\chi\) on generators for \(\left(\mathbb{Z}/2200\mathbb{Z}\right)^\times\).

\(n\)	\(177\)	\(551\)	\(1101\)	\(1201\)
\(\chi(n)\)	\(1\)	\(1\)	\(-1\)	\(-1\)

Coefficient data

For each \(n\) we display the coefficients of the \(q\)-expansion \(a_n\), the Satake parameters \(\alpha_p\), and the Satake angles \(\theta_p = \textrm{Arg}(\alpha_p)\).

(See \(a_n\) instead)

\(p\)	\(a_p\)	\(a_p / p^{(k-1)/2}\)	\( \alpha_p\)			\( \theta_p \)
\(2\)	−1.00000	−1.00000
			\(3\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(5\)	0	0
			\(7\)	0	0	1.00000			\(0\)
−1.00000						\(\pi\)
\(11\)	1.00000	1.00000
			\(13\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
0.500000	+	0.866025i				\(0.333333\pi\)
\(17\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(19\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(23\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(29\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(31\)	−1.00000	−1.00000	−0.500000	−	0.866025i	\(-0.666667\pi\)
			−0.500000	+	0.866025i	\(0.666667\pi\)
\(37\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(41\)	0	0	1.00000			\(0\)
			−1.00000			\(\pi\)
\(43\)	1.00000	1.00000	0.500000	−	0.866025i	\(-0.333333\pi\)
			0.500000	+	0.866025i	\(0.333333\pi\)
\(47\)	−2.00000	−2.00000	−1.00000			\(\pi\)
			−1.00000			\(\pi\)

(See \(a_n\) instead)

Twists

By twisting character
Char	Parity	Ord	Type	Twist	Min	Dim
1.1	even	1	trivial	2200.1.d.b.901.1	yes	1
5.2	odd	4		2200.1.o.b.549.1		2
5.3	odd	4		2200.1.o.b.549.2		2
5.4	even	2		2200.1.d.d.901.1	yes	1
8.5	even	2		2200.1.d.c.901.1	yes	1
11.10	odd	2		2200.1.d.c.901.1	yes	1
40.13	odd	4		2200.1.o.a.549.1		2
40.29	even	2		2200.1.d.a.901.1	✓	1
40.37	odd	4		2200.1.o.a.549.2		2
55.32	even	4		2200.1.o.a.549.2		2
55.43	even	4		2200.1.o.a.549.1		2
55.54	odd	2		2200.1.d.a.901.1	✓	1
88.21	odd	2	CM	2200.1.d.b.901.1	yes	1
440.109	odd	2		2200.1.d.d.901.1	yes	1
440.197	even	4		2200.1.o.b.549.1		2
440.373	even	4		2200.1.o.b.549.2		2

    

By twisted newform
Twist	Min	Dim	Char	Parity	Ord	Type
2200.1.d.a.901.1	✓	1	40.29	even	2
2200.1.d.a.901.1	✓	1	55.54	odd	2
2200.1.d.b.901.1	yes	1	1.1	even	1	trivial
2200.1.d.b.901.1	yes	1	88.21	odd	2	CM
2200.1.d.c.901.1	yes	1	8.5	even	2
2200.1.d.c.901.1	yes	1	11.10	odd	2
2200.1.d.d.901.1	yes	1	5.4	even	2
2200.1.d.d.901.1	yes	1	440.109	odd	2
2200.1.o.a.549.1		2	40.13	odd	4
2200.1.o.a.549.1		2	55.43	even	4
2200.1.o.a.549.2		2	40.37	odd	4
2200.1.o.a.549.2		2	55.32	even	4
2200.1.o.b.549.1		2	5.2	odd	4
2200.1.o.b.549.1		2	440.197	even	4
2200.1.o.b.549.2		2	5.3	odd	4
2200.1.o.b.549.2		2	440.373	even	4